Mathematics Concept & Skill Development Lecture Series:
Webvideo consolidation of site
lessons and lesson ideas in preparation. Price to be determined.

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Are you a careful reader, writer and thinker?Five logic chapters lead to greater precision and comprehension in reading and
writing at home, in school, at work and in mathematics. -
1 versus 2-way implication rules - A different starting point - Writing or introducting
the 1-way implication rule IF B THEN A as A IF B may emphasize
the difference between it or the latter, and the 2-way implication A IF and ONLY IF B.
-
Deductive Chains of Reason - See which implications can and cannot be used together
to arrive at more implications or conclusions,
-
Mathematical Induction - a light romantic view that becomes serious. -
Responsibility Arguments - his, hers or no one's -
Islands and Divisions of Knowledge - a model for many arts and
disciplines including mathematics course design: Different entry
points may make learning and teaching easier. Are you ready for them?

Deciml Place Value - funny ways to read multidigit decimals forwards and
backwards in groups of 3 or 6. -
Decimals for Tutors - lean how to explain or justify operations.
Long division of polynomials is easier for student who master long
division with decimals. -
Primes Factors - Efficient fraction skills and later studies of
polynomials depend on this. -
Fractions + Ratios - See how raising terms to obtain equivalent fractions leads to methods for
addition, comparison, subtraction, multiplication and division of
fractions. -
Arithmetic with units - Skills of value in daily life and in the
further study of rates, proportionality constants and computations in
science & technology.

What is
a Variable? - this entertaining oral & geometric view
may be before and besides more formal definitions - is the view mathematically
correct? -
Formula Evaluation - Seeing and showing how to do and
record steps or intermediate results of multistep methods allows the
steps or results to be seen and checked as done or later; and will
improve both marks and skill. The format here
allows the domino effects of care and the domino effects of mistakes
to be seen. It also emphasizes a proper use of the equal sign. -
Solve
Linear Eqns with & then without fractional operations on line segments - meet an visual introduction and learn how to
present do and record steps in a way that demonstrate skill; learn
how to check answers, set the stage for solving word problems by
by learning how to solve systems of equations in essentially one
unknown, set the stage for solving triangular and general systems of
equations algebraically. -
Function notation for Computation Rules - another way of looking
at formulas. Does a computation rule, and any rule equivalent to it, define a function? -
Axioms [some] as equivalent Computation Rule view - another way for understanding
and explaining axioms. -
Using
Formulas Backwards - Most rules, formulas and relations may be used forwards and backwards.
Talking about it should lead everyone
to expect a backward use alone or plural, after mastery of forward use. Proportionality
relations may be use backward first to find a proportionality constant before being
used forwards and backwards to solve a problem.

Welcome. The leading elements of Online books Three
Skills For Algebra, and Why
Slopes and More Math. stem from early college lessons that amused and
informed recent high school students, science, engineering and education
students, and adults in remedial to advanced mathematics lessons.

Three
Skills For Algebra begins with logic to test and sharpen
reading and writing, continues with a high school level arithmetic review
exercises to find gaps in solution writing skills, and continues
again with algebra review chapters to strengthen algebraic abilities. The chapters and exercises were
written to fill gaps in their senior high schoool command of mathematics

The leading elements of
Why Slopes and More Math gives a light technical context for the
senior high school study of slopes, of factored polynomials and of end-point and
interior maxima and minima of functions through sign analysis of formulas
for slopes and derivatives of functions, formulas given in factored form.
Thus leading elements of online book
Why Slopes and More Math begins with the algebraically easy part of a first course in
calculus, the part that comes after algebraically harder
decussion of limits, continuity and convergence.

The leading elements of both books together show how different starting points make calculus and senior highs chool mathematics easier.

"Would you tell me, please, which way I ought to go from
here?"
"That depends a good deal on where you want to get to," said the Cat.
"I don't much care where--" said Alice.
"Then it doesn't matter which way you go," said the Cat.
"--so long as I get SOMEWHERE," Alice added as an explanation.
"Oh, you're sure to do that," said the Cat, "if you only walk long
enough." (Alice's Adventures in Wonderland, Chapter 6)
--

Site lesson and lesson ideas in implying step by step a stronger
and more effective path for direct instruction may mute or lessen the
claims in education physchology or constructivism 1990 onward that direct
instruction does not provide an effective way to teach mathematics - the
fears and troubles of too many or most being evidence of that.

The question why learn or study a subject
or a topic appears is a sign of intelligence. Some
students have parents who say mathematics mastery is important. But many
have parents who in recalling their experience express a dislike for
mathematics after primary school. But if we combine ends and values from
earlier times, we may arrive at overlapping sets of ends and values for
learning and teaching primary and high school mathematics. The first ends reflect the actual or potential needs of
adult or daily life, and in trades and activities that do not require
common studies. The third end reflect the needs of calculus-based college
programs and of advanced, senior high school science courses. The first
two ends are more immediate than the third end. For the first two ends,
if not the third, over-preparation is better than under-preparation to
students and their families earn their livelihoods and to rationally
defend their interests in a world where daily behaviour, and contracts
involving money matters or income have huge consequences for individuals
and their families.

For mathematics and logic instruction, preparing children and teenagers
to earn income as adults may meet employers needs but more
importantly, it may meet the needs of students and their
families for income from employment or self-employment, and
defending their own interests while changing jobs or being fired. High school, trade school, undergraduate
university programs and graduate university programs may open doors for
worthwhile employment, but depending on economic times, education too long may also distract
from gainful or worthwhile work. There are no simple answers. Where does education for the sake of students begin and
where does it end? It may begin by showing students early how to handle money
matters in daily and adult life from not going into debt while buying
or selling to evaluating the immediate or long-term value of a
mortgage, a pension plan or the income stream and benefits of a job
with or without benefits may help them face or avoid common situations
and difficulties.

Early mathematics skill development in primary school and junior high school may provide common arithmetic and
geometric needs in daily and adult life. That may include say the
common needs of precollege trades and professions. Preparation for
daily or adult life at home, at work and in travel requires us to
count, measure and calculate with money, time, length, area, volume,
speed and rates of change on paper or with the geometric help of
maps, plans and diagrams carefully drawn to scale. Arithmetic mastery
may include formula evaluation. Early skill development should make
us want to avoid the domino effects of errors. That has value for all
multistep methods in- and out-side of mathematics. Early skill
development, well done, may make mastery of routine skills and
concept common, while providing a partial base for college studies.
Focus mostly on method and ideas with actual and then take-home value
may lead students and their families to value and want mathematics
and logic in early instruction.

The scout motto "be prepared" for what may come applies. For better
and worse, numerical and logical skills and concepts are needed in
daily and adult life to understand others, to read and write
instructions precisely, and to correct yourself or others. There is
a great risk of making incorrect decisions if you do not fully
understand the numerical and logic reasoning used in arguments and
agreements between yourself and others. Mastery of logic and basic
mathematics, the more the better, will help you quietly recognize
faulty decision making, yours or that of others.

In early or besides mastery ofmathematics, logic mastery leads to more or full precision in
reading, writing, speaking and listening. More or full precision will ease or
avoid confusion in following and giving instructions in many arts and
disciplines at home, in school and in the workplace. Logic mastery
sooner rather than later is best for its take-home value for studies and then wokk. But when
may depend on each student. Before or beside logic mastery, early
skill mathematics or quantitative skill development may emphasize how to do and record measurement and
arithmetic steps precisely, so that the steps can be seen and
verified, and so that students become aware of the need to avoid the
falling domino effect of errors. In this falling domino effect, a
mistake in one step leads the following steps to being in error,
except in the lucky case where a second or further mistake cancels
the effect of the earlier ones. For that, there should be no credit.
Plug: The leading math-free chapters of online Volume Three
Skills for Algebra on implication rules and their use in
deductive reason may lead the not too young to logic mastery.

Mid- and senior- high school mathematics and logic skill development
may build on earlier development to serve the needs of senior high
school science and technology courses, and the needs of
calculus-based college programs in commerce, science, engineering and
technology. Calculus in the first instance consists of calculation of
slopes for linear and nonlinear curvers y =f(x). The key role of
slopes in calculus explains why slopes and rates of change need to be
introduced earlier studies. Hint, Hint Site volume 3 with its
light calculus previews offer a context for the study of slopes,
factored polynomials and function maxima and minima may amuse and
inform students in courses leading to calculus and in the first weeks
of calculus.

Students heading for calculus-based, college programs in business, if
they avoid demanding high school science courses, will not see senior
high school mathematics used before arriving in college. To compensate,
long-term value needs to be emphasized. Emphasize how the calculations and logic of
college level programs requiring calculus will be more difficult to follow or use
and bend to our future requirements with a weak mastery of mathematics.
Site volumes 2 and 3 in forming and reforming the views of students and
teachers in senior high school mathematics as indicated above may inform
and amuse, and in the process provide some context and motivation for the
study of slopes, factored polynomials, function maxima and minima, and
calculus too.

All the ideas described briefly below are explained in more detail in
site algebra starter lessons and in site Volume 2, Three Skills and
Algebra. The arithmetic related ideas could have been placed with site
arithmetic lessons instead.

Arithmetic and algebraic expressions are often to complicated to
read aloud, term term by term. Diagrams too are better seen than "read
aloud". Outside of mathematics, a picture is worth a thousand words. In
mathematics, a symbol, an expression or a diagram better seen and
grasped in silence may also be worth a hundred to a thousand words.
There has been a great silence in arithmetic, algebra, geometry and
calculus because mathematical ideas and methods are often better
written and drawn in silence instead of being expressed and explained
aloud. Yet we may deliberately use more words to introduce skills and
concepts clearer, to talking unifying themes, and to improve
communication in circumstances where writing or drawing is not
possible. While demonstration how appears in site material, we will
identify where the greater use of words is possible. There is more to
mathematics than be given a formula, and numbers to use in it. But
remember, pictures and diagrams too can be employed alone and besides
words to make skills and concepts easier to learn and teach.

Before and besides the role of letters and symbols in algebra, we may use
words and numerical examples to talk about about and show how to
calculate totals and products by adding and multipling subtotals and
subproducts. We may also talk informally but precisely about counts and
measures as being known or not, constant or not, forgotten or not, and
variable or not. Many technical terms may be introduced and understood
before and besides the letters and symbols. Moreover, to gossip or talk
about people, places and activities, we need names, labels and phrases to
identify them. In mathematics, names and descriptive phrases such as the
compound growth formula, the rectangular area calculation, the
distributive law and the Chinese Square Proof of the Pythagorean Theorem
allow us to gossip and talk about calculations and further ideas in
situations where symbols and diagrams cannot be formed nor read. Most
formulas, methods and practices in mathematics and logic are named. For
people wanting and able to talk about what they learning with others,
learning the names becomes an asset and not a burden.

In describing how to calculate averages and how to compute the
perimeter of a polygon, word descriptions of how may be simpler or
not to understand and explain than formulas. As a first example,
the average of a set or sequence of numbers is given by their total
divided divided by the number (count) of set or sequence elements. As a
second example, the perimeter of a polygon is given by the sum of the
lengths of it sides, or more briefly by the instruction: add the sides.
As a third example, the total area of a region consisting of
non-overlapping subregions is given by a sum of subareas. In early
mathematics instruction, how to compute this or that may be easier to
understand and explain with words with the use of letters or symbols
being more complicated. But for the compound interest or growth
formula, for the quadratic formulas and later for the chain rule - do
not worry what computations these phrases name or identify, the the
letters and symbols in them are worth a thousand words. The greater use
of words advocated for earlier instruction here is not possible in
later instruction. So the silence will return.

Using rules and formulas forwards and backward, and talking about it
may end a further silence. Talking and writing about the forward and
backward use of rules and formulas provides a unifying verbal theme for
the study of logic, mathematics and science in school and college
studies. Most if not all rules and formulas are not only used directly in
a forward sense but also indirectly or backwards. Determing the constant
in a proportionality relation uses the relationship, an equation,
backwards. Once it is found, the proportionality relations may then be
used or rewritten forwards and backwards to compute or express the value
of one number or quantity in terms of others. The example here may not be
familar to you if you have not seen them, but by talking about the
forward and backward use of rules, formulas and proportionality
relations, the backward use will be expected and not be another surprise
for students weak and strong of mathematics, logic and science. This
forwards and backwards use is common pattern previously met and mastered
case by case in silence. Talking and writing about it introduces or
extends the oral dimension of skill and concept development.

Site algebra starter lessons and the online chapters of Volume 2, Three
Skills for Algebra, material, show how to learn and teach skills and
concepts with words, forwards and backwards. Algebra starter lessons
include a geometric, stick diagram introduction for solving linear
equations in a way that visually proves or improves fraction skills and
sense. Here fractional operations on stick diagrams are suppose to make
the algebraic solution of linear equations easier to grasp. However, in
entertaining a group of students during a one hour, substitute teaching
assignment, one keen student could not make the transition from solving
with stick diagrams to solving algebraically. It was not my place to give
him extra instruction. He may have been better served by more stick
diagram examples, or by a leap to the algebraic method. I cannot say.

Geometry too can help with the introduction of calculus and in providing
motivation or context for the study of slopes (remember the domain name
is whyslopes.com) and the study of factored polynomials alone and in
ratios (rational expressions). See site Volume 3, Why Slopes and More
Mathematics, online in full with a fall 1983 why slopes prequel. Volume 3
in a preview of calculus provide geometric motivation for the study of
slopes and factored polynomial to the location of maxima and minima of
functions.

The site introduction of complex numbers is geometric instead of
algebraic. It follows or re-invents a path in a 1951 book on Secondary
Mathematics (possibilities) by Howard Fuhr, a mathematician who
masqueraded as a mathematics education professor at Columbia University
and who as part of the NCTM leadership in the 1960s help develop and
implement the college-oriented Modern Mathematics Programs for skill and
concept development in primary and high school mathematics from counting
to calculus. The level of rigour in this geometric introduction of
complex numbers is not less than that in the geometric introduction of
trigonometry using triangles and/or the unit circles drawn in a Cartesian
plane.

The big steps in modern mathematics programs were too hard for
many to follow. Site material offer smaller steps to compensate. Before
modern mathematics programs, instruction had a greater focus on skills
and concepts with value for daily and adult life - work, mortgages and
investments included. The discussion of ends and values above suggests
preparation for daily and adult life as much as possible first and
foremost, and on preparation for college second while emphasing
anything in the latter that could have take-home value.

Site composition was driven by a search to remedy the skill and concept
development difficulties stemming from steps too big and steps without
clear value for students and their families in earlier programs in
mathematics and logic education - programs which aimed for student
mastery of selected skills and concepts. In consequence, site lessons and
lesson ideas include many expositional innovations to aid skill and
concept development. Most, if not all, are mathematically correct, with a
few small departures from earlier views to make instruction simpler.

In calculus and secondary mathematics, late primary mathematics too,
there are many different starting points for instruction.

For example, the site development of prime numbers begins with a
definition that is not the most general but with a definition that is
likely the easiest for students to understand and apply. For a second
example, the site essay on what is a variable, by talking or writing
about numbers and quantities varying in one sense or another, we
provide a prequel to the later, more formal and more algebraically
advanced view of what is a variable, a prequel that is easily
understood because it is wordy and pre-algebraic. For more examples,
see the site geometric development of complex numbers before trig, and
see light calculus preview in Volume 3, Why Slopes and More Math, and
see, still in Volume 3, the decimal prequel to the epsilon-delta view
or definitions of limits and continuity.

The choice of starting point need not reflect the conventions of higher
mathematics, conventions that may be arbitrary despited being widely
accepted. Instead, the choice of starting point may reflect the objective
of making skill and concept development simpler for students and their
teachers. The harder starting points may be left to advanced studies
involving fewer students and teachers.

Mathematics Literacy: Since students may leave school early,
we need to show them and give them mastery of reading, writing,
arithmetic and geometry with actual or potential take-home value for
their daily and adult life in local and distant communities. While
learning mathematics with comprehension is best, the take-home value
of basic and routine skills needed for daily and adult to important
to insist upon mastery with comprehension. In this course design and
delivery should emphasize the domino effect of errors in multistep
methods, numerical or geometric. And in arithmetic, students should
be shown how to do and record steps in a manner that their skills can
be seen and checked as done or later. In practical, skill-based arts
and disciplines from cooking to mathematics, skills needs to be
demonstrated to be believed, and indeed to be both taught and
mastered. In general, there are too many skills for a student to find
them or their refined form by discovery. The challenge for early
mathematics instruction is to identify and provide observable and
thus verifiable skills with take-home value that serves common or
routine needs while seamlessly preparing students for late
instruction.

Geometry with Proportionality First: To quickly support the
common, actual or potential, geometric use of maps, plans and
diagrams drawn to scale in daily and adult life, and in precollege
trades and professions, the site webvideo exposition of geometry may
include SAS, ASA and SSS methods or practices for the construction of
similar or proportional triangles, and in general assume that in
maps, plans and diagrams drawn to different scales that corresponding
angles are equal and corresponding lengths are proportional. So the
similarity or proportionality present in maps, plans and diagrams
drawn to scale may be exploited to indirectly measure angles and
lengths, and quantities computed from them. Trigonometry may then be
introduced as a way to calculate angles and lengths instead of
obtaining them direct from measurements, actual or of the
corresponding angles and lengths on maps, plans and diagrams drawn to
scale. The early mastery of common, and easily understood and
repreated practices with maps, plans and diagrams drawn to scale
provides a context for and even implies the assumptions and axioms of
Euclidean Geometry.

Geometry with Congruence or Isometery Second: For simpler or
more accessible account of Euclidean geometry, the site account does
not include a proof of the Pythagorean thereom. The Chinese Square
Dissection proof provides a more accessible alternative. The latter
is presented online in Volume 2, Three Skills for Algebra. Without
the Pythagorean thereom, Euclidean geometry may be easy enough to
return to the North American classroom in a way that shows high
school or college students how logic in the form of implication rules
alone and in direct deductive chains of reason appears in
mathematics.

Counting and Arithmetic with Decimals: Decimal place value is
the key to counting. We assume every set of fewer than 10, 100, 1000
and 10000, etc, can be divided into a group of upto 9 units, a group
upto 9 groups of ten, upto 9 groups of 100 and upto 9 groups of 1000
in manner that the count between 0 and 9 of units, 10s, 100s and
1000s are unique, albeit the division of set elements into groups of
units, 10s, 100s and 1000s is not unique. The foregoing division or
partition gives a unique, multidigit decimal way to write and record
the count or number of set elements in which each unit has a place
value. The concept of place value leads to and easily justifies
arithmetic counting shortcuts involving the addition, comparision,
subtraction and multiplication and even division of decimals. The
details are given in the site arithmetics section along with North
American, metric (or SI) and UK-German methods for writing and
reading aloud with words multidigit decimals without and then with a
decimal point. Comprehension of operations with decimals enriches
early instruction and may help some master these operations. Others,
most others perhaps, may find full explanation of why some operations
work too complicated for their liking. For them skill and confidence
in decimal methods may follow learning how to use the methods to
obtain repeatable and reproducible results via steps observable and,
if need-be correctable.

Counting and Arithmetic with Fractions: The fraction three
quarters when written or read aloud means three times a quarter. A
quarter œ is a unit fraction. Proper and improper fractions with the
same denominator all give a number or count of a unit fraction, that
associated with the same denominator. With the aid of decimal
representations forms of numerators, it is an easy matter to count,
add, compare, subtract and even divide multiples of a single unit
fraction. It also an easy matter to multiply a multiple of a single
fraction by a whole number - to form a multiple of a multiple. By
long division and regrouping, each improper fractions is equivalent
to a mixed numbers. In primary and secondary school, students may be
shown how to add and subtract fractions with unlike denominators by
raising terms to convert each fraction to another equivalent
fraction, so after conversion, each has a common denominator and so
is a multiple of a common unit fraction. Following this, students may
be shown how to compare, multiple and divide fractions by rote. Site
fraction lessons in contrast show how raising terms to obtain like
denominators explains and justifies methods to compare and divide
fractions while raising terms to ensure the numerator of the
multiplicand is a multiple of the denominator of the multiplier
explains and justify methods for fraction multiplication. The
justification of arithmetic with fractions sets the stage for the
justification of arithmetic with decimal fractions (multiples of
one-tenth, one hundredth, one thousandths) that usually denoted by
multidigit decimals with digits after and even before a decimal
point.

Comprehension of operations with fractions agains enriches early
instruction and may help some master these operations. Others, most
others perhaps, may find full explanation of why some operations work
too complicated for their liking. For them skill and confidence in
decimal methods may follow learning how to use the methods to obtain
repeatable and reproducible results via steps observable and, if
need-be correctable.

Prime Numbers and Fractions: For algebra alone or as part of
calculus, and for operations with complex numbers, students need an
efficient command of arithmetic with fractions where the denominators
are say less than 200. Prime factorization of whole numbers less than
200 is useful here. The development of prime number factorization
methods in the site arithmetic section shows how to use time tables
to recognize small primes, and how to use an olde square rule method
to quickly and efficiently obtain prime number factorization of whole
number less than 289 = 172, and to recognize primes less
than 289 as well. The foregoing path as demostrated in site
arithmetic section may be easier for people to learn and teach.
Prime factorization is also useful for a "simplification" of roots
involving whole numbers or their fractions, a simplification often
seen in trig and calculus. Mastery of exact arithmetic in high
mathematics requires mastery of some cosmetic standards or
conventions for the expression of fractions, roots and radicals.

Arithmetic with units and denominate Numbers - missing. Units
of measure and counting appear directly in daily and adult life, and
also in science and technology. Units of measure also appear in the
description of speed, acceleration and other first and second order
rates - rates that may be described as derivatives in calculus.
Modern mathematics programs did not mention nor sanction the use of
units and their multiples (denominate numbers) in high school and
college studies, albeit this use appear in science courses and in
some practical examples met in mathematics courses in trigonometry
and before. The site account of arithmetic and fractions with units
compensates for this. Albeit, the compensation is given in a do this,
do that manner, because of a lack of words on my part to provide
greater comprehension. Readers are invited to provide remedies. Early
algebra courses today may introduce monomials (products of letters or
"variables" to various powers) and operations on them alone and in
fractions before students understand the computational significance
of monomials and operations on them. Site algebra starter lessons
explanation of equivalent computation rules may provide a remedy for
that. But before or besides algebra, The same exercises with
monomials given by numerical multiple of products of units to various
powers may be more meaningful to students, while be a prerequisite to
the numerical description of rates and proportionality constants.

Algebra Starter Lessons. Showing students how to do and record
numerical and algebraic steps in ways that can be seen and checked when
done or later makes their mastery of multistep methods observable, and
hence verifiable or correctable. Showing should also make students
aware of the domino effects of mistakes, and the care needed to avoid
or correct such errors. The introduction and assumption of methods to
compute totals and products using subtotals and subproducts employs
practices that are too complicated in high school instruction to derive
from the usual axioms for arithmetic with real numbers. But the
assumption of these methods or practices extends the usual axioms and
from the perspective of advance mathematics gives a very redundant set
of axioms. But the same redundancy is justified as it makes early
instruction easier and more effective, and the extra assumptions have
immediate take-home value for daily and adult life not present in the
usual axioms. Now the usual axioms are best understood besides or after
a math-free mastery of logic. The usual axioms for the distributive,
commutative and associative law are algebraically described. Many
students find the algebraic description too remote or abstract. But if
we introduce the concept that each algebraic expression give a unique
computation rule, one that that be evaluated on paper or with the aid
of a program on a calculator or computer, we may observe from numerical
examples that different computation rules appear to be equivalent in
the sense that they give the same result. This small step of
introducing the concept of equivalent computation rules provides
another context, a different starting point, for understanding and
explaining distributive, commutative and associative laws in arithmetic
with many kinds of numbers, and eventually with numbers being replaced
by computation rules - those with numerical values.

Arithmetic without Calculators: To be over-prepared is better
and less risky than being under-prepared. A written, calculator-free
mastery of arithmetic with signs, decimals, fractions; with units of
measures; and with number theory practices is needed for a full,
traditional, mastery of algebra, trigonometry, complex numbers and
calculus. A full mastery of arithmetic with units of counting and
measures also has value for adult and daily life, and for further
studies in commerce, science, engineering and even mathematics
itself.

In modern urban life we depend on machines to simplify our daily
life. But calculators usage both simplifies and weakens mathematics
mastery, or that needed to understand decimals, fractions, algebra,
trigonometry and calculus. As a master of my subject with standards
for skill and concept development, I see the student who can only do
arithmetic with the aid of a calculator as being handicapped from
being too spoilt in earlier instruction. Any expectation that
quantitative skills and disciplines can be well-taught without a
written mastery of arithmetic with decimals and fractions is false.

Again, manually learning how to do and record work in steps that can
be seen and corrected as done or later may begin with evaluation of
arithmetic expressions and algebraic formulas. While calculators are
useful, failure to require manual student mastery of arithmetic
removes a starting point for observable skill and concept
development. In particular, mastery of observable steps that can
be seen and confirmed or corrected as done or later is also is key
part of showing and demonstrating abilities in problem solving, in
writing proofs and employing multistep methods at home, at work and
in studies in many arts and disciplines.

More on Ends and Values

Mathematics after primary school has been difficult and without immediate
value for one generation after another. While some students have parents
who did well or who encourage skill and concept development, other
students have parents who may say mathematics after arithmetic is a waste
of time. High school and college students may attend courses because
those courses are required. In high school and college, students who base
their efforts only on whether or not their teachers are pleasing have a
shallow context and motivation for learning. Students for whom doing well
in tests and finals is the only motivation also have shallow reason for
learning. Cultural values for learning may appeal to some. But practical
ends and values may appeal to more.

In primary school, students and their families may see the 4Rs (reading,
writing, reasoning and arithmetic) as being useful in adult and daily
life. There-in lies content and motivation. But at the junior- and
mid-high school level, some mathematics and logic lessons are of actual
or potential service to daily and adult life for decision-making and
money-matters at home and the work place. Other lessons only have
long-term value for college programs that some students may never enter
or complete. Instruction may lean to the first kind of lessons initially
to provide ends and values easily understood and appreciated by students
and their families. Emphasizing the more useful methods and concepts
first may help retain student motivation and also help those who have
leave school early. But eventually, high school and college mathematics
has less and less take-home value besides more and more value for future
studies or courses that students may not see. Here again, instruction may
focus on the take-home value, when present to provide motivation.

At the precalculus level, instruction should focus on two kinds of skills
and concepts, those that have actual or potential take-home value for
daily & adult life, and for precollege trades and activities; and
those that prepare students for a light and then deeper command of
calculus. In the former, I would include a set-based development of
probability theory. In both streams, I might include matrix operations
but not linear programming. The latter can be left to college programs in
commerce, science, engineering and technology. I would restrict high
school mathematics to computations and proofs that are lead to repeatable
and reproducible results, and to the computation of averages useful in
small business for estimating demand for products and services being
sold. Further elements of descriptive statistics, I would leave to
college studies, or to high school courses on critical thinking.

The recommended focus may mean fewer topics are taught. For students not
heading for calculus-based studies, less with a focus of skills and
concepts with take-home value may be best. In the preparation of students
for calculus and senior high school mathematics, multiple topics with no
short-term value may be met. That short-term value will vary between
students. Students in courses required to prepare for calculus who do
take mathematically demanding, senior high school courses will see more
short-term value. In general, calculus and preparation for calculus is a
long demanding path which many find difficult or hard to complete. But,
here is a plug, site Volumes 2 and 3, make the path easier and throught
calculus preview make calculus and precalculus easier and more appealing.

To serve the skill and concept needs of the common person in the street,
we need to put first those skills and concepts with actual or potential
value for daily and adult life. Then students may attend school and go
home with methods that help themselves or their families in money and
other matters. Near the end of school coverage of arithmetic, geometric
and logic (or reading and writing) skills and concepts with actual or
potential service for daily and adult life, more algebra and higher level
geometry skills may be introduced to revisit and reinforce the foregoing
service while being of service to more trades and activities at the
precalculus level, and also being of service or preparation for senior
high school science courses and perhaps later studies in calculus. The
multiple ends and values in the foregoing need to be balanced. The
balance may depend on the local or immediate needs of students and their
families, that is, how long students are likely to remain in school; on
whether or not, they are likely to see all all ends and values served;
and on whether or not, the students are quick or slow learners.

The concept and skill development standards and principles for
instruction in results-oriented arts and disciplines, as espoused in site
material, seek to provide students with an observable and verifiable
know-how of the ideas and methods currently forming and characterizing
each art or discipline. The latter presents a moving targets as best
practices in each may vary over time and place. But in a moving target,
concept and skill mastery may be seen or empirically measured by student
response to questions. In each such art and discipline, students are
expected to retain know-how and build on it in a progressive manner, with
regression being a sign of weakness, or absence too long from practice in
an art or discipline. Each art or discipline comes with different
cultural and practical values, some more important than others in ways
that may justify its instruction or not in each school or school system.

Morover, course design and delivery needs to acknowledge that there are
multiple intelligences in learning and teaching styles. A style that is
suitable for instruction in the humanities where conclusions are highly
subjective is not suitable for instruction in mathematics and science
where the benefits, origins and limitations of ideas and methods should
be shared.

In modern mathematics programs for secondary mathematics
education, direct instruction aimed at student mastery of given
concepts and skills has been uncertain and unreliable due to steps too
big or hard for most to follow, and due a college-oriented choice of
concepts and skills with value too long-term for students and their
families. Those steps too big undermined course design and delivery.
However, direct instruction can address its own problems by serving
short- and long-term ends and values in the selection and arrangement
of course topics, and in offering smaller, more accessible and reliable
steps for concept and skill mastery. The key question is whether or not
remedies based on the smaller and alternative steps in site lessons and lesson ideas,
alone or with the proposed ends and values above, will be
effective..

Site lessons and lessons ideas from counting to calculus provide a
foundation for college level studies of modern mathematics. Site lessons
and lesson ideas offer student and their teachers a mastery of concepts
and skills with comprehension, based on a redundant
set of practices and axioms, whose redundancy can be explained and
removed in college course in or leading to modern mathematics. The ends
and value further offer reasons for mathematics and logic mastery that
students and their families are more likely to appreciate before
preparation for calculus becomes the main focus of instruction at the
senior high school level. For calculus, Chapter 14 of site Volume 3, Why
Slopes and More Mathematics, offers a decimal, error control development
of limit and continuity concepts that may stand alone, or be used to make
the epsilon-delta development much easier to understand and explain. Site
departures in early instruction from modern mathematics are intended to
provide TCPITS an more accessible view, but they are also intended to
develop the logical and algebraic maturity needed for college and senior
high school students to study modern mathematics if they choose or where
it appears in their programs of study.

Indirect Instruction Benefits and Limits

Indirect instruction in mathematics has the advantage of enriching skills and concept
mastery in classes where there is time for individual and group
creativity, and where teachers not all trained in mathematics are shown how to provide problems and
circumstances which studentsmay investigate to discover or build
their own ccmprehension. But with or where teachers are not fully versed in mathematics or
course content, it appears far simpler to provide instructors with lessons easily understood
and repeated in class, which avoid the affects o steps too large, not for all, but for
most to follow. For example, the verbal introduction of algebra in Volume 2, Three Skills For
Algebra, and the first six or seven chapters of Volume 3, Why Slopes and More Mathematics, provide
lessons and lesson idess, easily understood and repeated in class, and in the process may introduce
learning difficulties of students with instructors at many levels. Here advocates of indirect
instruction while declaring student mastery of given skills and concepts in direct instruction
to be a substandard objective for mathematics education essentially kept the course design and content
which direct instruction employed in its identification of skills and concepts for student mastery. Moreover,
educational authorities at the precollege level retain final examinations for the yearly end of most high school
mathematics courses. Final examination by their very nature test student mastery of chosen skill
and concepts. Due to the continued presence of mathematics final examinations which tests student
mastery of given skills and concept, fairness in student evaluation requires all the given
skills and concepts be clearly explain, illustrated and checked first in class. So direct instruction
is still required. For fairness, student mastery of given skills and concepts requires both teacher and
student awareness of how later skills and concepts stand on earlier ones. Otherwise, students will be promoted
with the necessary background to succeed.

When and where direct instruction has clear steps or lessons to
provide student mastery of important skills and concepts, teachers and course
designers may
provide circumstances and pose questions to indirectly lead student to
formulate ideas and skills and gain the experience on which direct
instruction may stand. But where direct instruction lacks those clear
steps and lessons, it is doubtful that indirect instruction will
provide a practical and clearer path to to student mastery of the given
skills and concepts. The ability to explain matters directly is likely
a prerequisites to the ability to provide skill and concept mastery
indirectly.

Each program of instruction aim at mastery of given ideas and methods has
varying degrees of success and failure, and of motivation and alienation
for students and their families. In the case of modern mathematics
programs for secondary mathematics and calculus, the step by step
development was clear to some and due to the presence of steps too big,
not for but for some, confusing for others.
Site material in providing smaller steps allows steps too big to be
recognized and gives remedies - full or not - to be tried and tested.
Smaller steps should allow more to go
further.

Appendix - Closing Words

Still More on Ends and Values

In Canada and the USA five decades ago, that is, in the 1960s, modern
mathematics programs set forth the substance or content of high school
mathematics and calculus in a way that served the technical needs of
college bound students, those heading for university programs in
commerce, science and technology. Mathematical discussion leading to
those modern mathematics programs acknowledged the college-orientation
and mentioned that the needs of student not college bound were not being
addressed. However, for the sake of inclusivity, modern mathematics
programs were adopted for all instead of the just college bound. The
tradition of mathematics course design incomprehensible to parents -
different from what they had seen in school or college may have begun
then. From my perspective as a student and then as an instructor, the
modern and pre-modern mathematics course design in algebra at least
included steps too large not for all, but for most to follow. In primary and secondary schools, the college-oriented selection of
content in modern mathematics programs helped move course design and
delivery away from serving the actual or potential needs of daily and
adult life, and in doing so offered less context and motivation for
mastery of given ideas and methods.

For senior high school mathematics, preparation for college programs of
the technical or mathematical kind provides one context and motivation
for college-bound students. Senior high school mathematics is of clear
service to students headed for science or engineering take senior
high school courses in physics, chemistry and biology. But students headed
for college commerce programs - those best studied with the help of
calculus - may or should be encourage to take senior high school courses
in science. Otherwise, physics-bsaed examples in senior high school
mathematics will have less value for them.

To provide context and
motivation for more students and their families, primary school and the leading years of high school may
deliberately develop mathematical and logical skills, concepts and work
habits with clear or identifiable actual or potential benefit in daily or
adult life. In particular, Primary school and the leading years of
secondary school may focus on basic skills in counting, figuring and
measuring of actual or potential service to daily and adult life.
Students and their families may value this focus. The leading years of
secondary school may further introduce and emphasize skill, concepts and
work habits of service in senior high school and college mathematics. In
this standards for mastery of given skills and concepts may be maintained
and supported by lesson and lesson ideas deliberately chosen and
continually reviewed to make the hard less so, without loss of substance;
and guided by the setting of final examinations set not by amateurs but
by subject experts. Here multilevel course design and delivery requires
knowledge of and respect of how later ideas and methods depend on earlier
ones. In this, while high school mathmatics as a whole may prepare
student for calculus-based college studies, the inclusion of skills and
concepts of service, actual or potential, to daily and adult life; money handling
included, or to
pre-college trades and professions could provide and sweeten the context
of secondary mathematics for all. And in the preparation for calculus, set formalism
may appear to set the stage for the later study perhaps of pure mathematics in college,
but in a minimal manner, one that does not overwhelm.

Play with this [unsigned]
Complex Number Java Applet
to visually do complex number arithmetic with polar and Cartesian coordinates and with the head-to-tail
addition of arrows in the plane. Click and drag complex numbers A and B to change their locations.

Pattern Based Reason

Online Volume 1A,
Pattern Based Reason, describes
origins, benefits and limits of rule- and pattern-based reason and decisions
in society, science, technology, engineering and mathematics. Not all is certain. We may strive for objectivity, but not
reach it. Online postscripts offer
a story-telling view of learning: [
A ] [
B ] [
C ] [
D ] to suggest how we share theory and practice in many fields of knowledge.

Site Reviews

1996 - Magellan, the McKinley
Internet Directory:

Mathphobics, this site may ease your fears of the subject, perhaps even
help you enjoy it. The tone of the little lessons and "appetizers" on
math and logic is unintimidating, sometimes funny and very clear. There
are a number of different angles offered, and you do not need to follow
any linear lesson plan. Just pick and peck. The site also offers some
reflections on teaching, so that teachers can not only use the site as
part of their lesson, but also learn from it.

... new sections on Complex Numbers and the Distributive Law
for Complex Numbers offer a short way to reach and explain:
trigonometry, the Pythagorean theorem,trig formulas for dot- and
cross-products, the cosine law,a converse to the Pythagorean Theorem

Math resources for both students and teachers are given on this site,
spanning the general topics of arithmetic, logic, algebra, calculus,
complex numbers, and Euclidean geometry. Lessons and how-tos with clear
descriptions of many important concepts provide a good foundation for
high school and college level mathematics. There are sample problems that
can help students prepare for exams, or teachers can make their own
assignments based on the problems. Everything presented on the site is
not only educational, but interesting as well. There is certainly plenty
of material; however, it is somewhat poorly organized. This does not take
away from the quality of the information, though.

... section Solving Linear Equations ... offers lesson ideas for
teaching linear equations in high school or college. The approach uses
stick diagrams to solve linear equations because they "provide a concrete
or visual context for many of the rules or patterns for solving
equations, a context that may develop equation solving skills and
confidence." The idea is to build up student confidence in problem
solving before presenting any formal algebraic statement of the rule and
patterns for solving equations. ...

-
Euclidean Geometry - See how chains of reason appears in and
besides geometric constructions. -
Complex Numbers - Learn how rectangular and polar coordinates may
be used for adding, multiplying and reflecting points in the plane,
in a manner known since the 1840s for representing and demystifying
"imaginary" numbers, and in a manner that provides a quicker,
mathematically correct, path for defining "circular" trigonometric
functions for all angles, not just acute ones, and easily obtaining
their properties. Students of vectors in the plane may appreciate the
complex number development of trig-formulas for dot- and
cross-products.
Lines-Slopes [I] - Take I & take II respectively assume no
knowledge and some knowledge of the tangent function in
trigonometry.

Why study slopes - this fall 1983 calculus appetizer shone in many
classes at the start of calculus. It could also be given after the intro of slopes
to introduce function maxima and minima at the ends of closed intervals. -
Why Factor Polynomials - Online Chapter 2 to 7 offer a light introduction function maxima
and minima while indicating why we calculate derivatives or slopes to linear and nonlinear
curves y =f(x) -
Arithmetic Exercises with hints of algebra. - Answers are given. If there are many
differences between your answers and those online, hire a tutor, one
has done very well in a full year of calculus to correct your work. You may be worse than you think.